Solving Equations: A Step-by-Step Guide
Hey math enthusiasts! Let's dive into solving equations, specifically the one we've got here: 5 - (3 - 2x) = 6x - 10 - x. Don't worry, it looks a bit intimidating at first, but trust me, with a few simple steps, we'll crack this code together. This isn't just about finding x; it's about understanding the process, so you can tackle any equation thrown your way. We'll break it down into manageable chunks, explaining each step along the way. Think of it like a treasure hunt, and x is the hidden treasure. Ready to find it? Let's go!
Step 1: Simplify Both Sides of the Equation
First things first, we need to tidy up both sides of the equation. This involves getting rid of parentheses and combining like terms. Let's start with the left side: 5 - (3 - 2x). Remember, the minus sign in front of the parentheses means we need to distribute the negative to both terms inside. So, the equation becomes 5 - 3 + 2x. Simplifying further, we get 2 + 2x. Awesome! We've made the left side much neater.
Now, let's turn our attention to the right side: 6x - 10 - x. Here, we can combine the x terms. 6x - x is the same as 5x. So, the right side simplifies to 5x - 10. Now, our equation looks much friendlier: 2 + 2x = 5x - 10. See? We're making progress. This initial simplification is a crucial step in solving any equation. It helps to isolate the variable and makes the subsequent steps much easier to handle. It's like preparing the ingredients before you start cooking – everything becomes smoother when things are in order. And remember, always double-check your signs and terms during this step to avoid any errors. Small mistakes can lead to big problems down the line, so take your time and be meticulous. Making sure you understand this part is key, and we can move on to the next. Great job, everyone!
Step 2: Isolate the Variable Terms
Now that both sides are simplified, it's time to gather all the x terms on one side of the equation. It doesn't matter which side you choose, but let's move the x terms to the right side in this case. We have 2 + 2x = 5x - 10. To move the 2x from the left side, we need to subtract 2x from both sides. This is based on the principle of maintaining the equation's balance – whatever you do to one side, you must do to the other. So, we get: 2 + 2x - 2x = 5x - 10 - 2x. The 2x terms on the left side cancel each other out, leaving us with just 2. On the right side, 5x - 2x becomes 3x. Therefore, the equation now looks like this: 2 = 3x - 10. We're getting closer to isolating x! This step is all about organizing the equation to make it easier to solve. Think of it as sorting your clothes – you put all the shirts together, all the pants together, and so on. Similarly, here we group all the terms containing the variable on one side. This makes the next step, where we isolate the variable, much more straightforward. Don't worry if it seems like a lot; with practice, it'll become second nature. Keep going, you're doing great!
Step 3: Isolate the Variable
Now we're on the home stretch! We have the equation 2 = 3x - 10. Our goal here is to isolate x completely. First, we need to get rid of the -10 on the right side. To do this, we'll add 10 to both sides of the equation. This gives us: 2 + 10 = 3x - 10 + 10. Simplifying this, we get: 12 = 3x. See how we're closing in on our treasure, x? Now, to isolate x, we need to get rid of the 3 that's multiplying it. We do this by dividing both sides of the equation by 3. So, we get: 12 / 3 = 3x / 3. This simplifies to: 4 = x. Or, if we prefer, x = 4. Huzzah! We've found the solution! This step is about removing any numbers that are added to or multiplied by the variable. You're effectively 'undoing' the operations that have been performed on the variable. By adding, subtracting, multiplying, or dividing both sides of the equation, you maintain the balance and eventually reveal the value of the variable. Remember, the key is to isolate the variable on one side of the equation. Once you achieve this, you've essentially solved the equation. Take a moment to appreciate the fruits of your labor! You've successfully navigated the equation maze and found the hidden treasure.
Step 4: Check Your Solution
It's always a good idea to check your solution to make sure it's correct. Let's substitute x = 4 back into the original equation: 5 - (3 - 2x) = 6x - 10 - x. Replacing x with 4, we get: 5 - (3 - 2 * 4) = 6 * 4 - 10 - 4. Now, let's simplify both sides. On the left side: 5 - (3 - 8) = 5 - (-5) = 5 + 5 = 10. On the right side: 24 - 10 - 4 = 14 - 4 = 10. Since both sides are equal (10 = 10), our solution is correct! Checking your solution is like proofreading your work. It's an essential step to catch any potential errors and ensure the accuracy of your answer. It provides you with confidence in your solution and allows you to catch any mistakes that you might have made during the solving process. Always take the time to substitute your answer back into the original equation and verify that both sides are balanced. This will not only confirm that your solution is correct, but also reinforce your understanding of the equation-solving process. Congrats, you are absolutely awesome!
Conclusion: Equation Solved!
So there you have it, guys! We've successfully solved the equation 5 - (3 - 2x) = 6x - 10 - x and found that x = 4. Remember, solving equations is a skill that improves with practice. The more equations you solve, the more comfortable and confident you'll become. Keep practicing, and don't be afraid to ask for help if you get stuck. Each equation you solve is a victory! Take pride in your accomplishments. If you ever feel overwhelmed, break the problem down into smaller steps, and always double-check your work. You are well on your way to becoming a math whiz! Congratulations on your hard work and dedication. Keep exploring the world of math, and remember that learning is a journey, not a destination. Embrace the challenges, celebrate your successes, and most importantly, have fun along the way. Until next time, keep solving and keep shining! You've got this!
